Bohr-Sommerfeld rules to all orders

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1 Bohr-Sommerfeld rules to all orders Yves Colin de Verdière September 2, 2004 Prépublication de l Institut Fourier n 652 (2004) Institut Fourier, Unité mixte de recherche CNRS-UJF 5582 BP 74, Saint Martin d Hères Cedex (France) yves.colin-de-verdiere@ujf-grenoble.fr ycolver/ 1 Introduction The goal of this paper is to give a rather simple algorithm which computes the Bohr-Sommerfeld quantization rules to all orders in the semi-classical parameter h for a semi-classical Hamiltonian Ĥ on the real line. The formula gives the high order terms in the expansion in powers of h of the semi-classical action using only integrals on the energy curves of quantities which are locally computable from the Weyl symbol. The recipe uses only the knowledge of the Moyal formula expressing the star product of Weyl symbols. It is important to note that our method assumes already the existence of Bohr-Sommerfeld rules to any order (which is usually shown using some precise Ansatz for the eigenfunctions, like the WKB-Maslov Ansatz) and the problem we adress here is only about ways to compute these corrections. Our way to get these high order corrections is inspired by A. Voros s thesis (1977) [9], [10]. The reference [1], where a very similar method is sketched, was given to us by A. Voros. We use also in an essential way the nice formula of Helffer-Sjöstrand expressing f(ĥ) in terms of the resolvent. 1 Keywords: Bohr-Sommerfeld rules, Moyal formula, functional calculus, pseudo-differential operator, spectral theory, quantization rules 2 MSC 2000: 34E05, 35P15, 35S99, 81Q10 1

2 2 The setting and the main result Let us give a smooth classical Hamiltonian H : T R R, where the symbol H admits the formal expansion H H 0 + hh h k H k + ; following [4] p.101, we will assume that H belongs to the space of symbols S o (m) for some order function m (for example m = (1 + ξ 2 ) p ) H + i is elliptic and define Ĥ = Op Weyl(H) with 3 Op Weyl (H)u(x) = e i(x y)ξ/h H( x + y, ξ)u(y) dydξ R 2 2πh. 2 The operator Ĥ is then essentially self-adjoint on L2 (R) with domain the Schwartz space S(R). In general, we will denote by σ Weyl (A) the Weyl symbol of the operator A. The hypothesis: We fix some compact intervall I = [E, E + ] R, E < E +, and we assume that there exists a topological ring A such that A = A A + with A ± a connected component of H0 1 (E ± ). We assume that H 0 has no critical point in A We assume that A is included in the disk bounded by A +. If it is not the case, we can always change H to H. We define the well W as the disk bounded by A +. Definition 1 Let H W : T R R be equal to H in W, > E + outside W and bounded. Then ĤW = Op Weyl (H W ) is a self-adjoint bounded operator. The semiclassical spectrum associated to the well W, denoted by σ W, is defined as follows: σ W = Spectrum(ĤW ) ], E + ]. The previous definition is usefull because σ W is independent of H W mod O(h ). Moreover, if H 1 0 (], E+ ]) = W 1 W N (connected components), then Spectrum(Ĥ) ], E+ ] = σ Wl + O(h ). 3 Contrary to the usual notation, we denote by dx 1 dx n the Lebesgue measure on R n in order to avoid confusions related to orientations problems. 2

3 The spectrum σ W [E, E + ] is then given mod O(h ) by the following Bohr- Sommerfeld rules S h (E n ) = 2πnh where n Z is the quantum number and the formal series S h (E) = S j (E)h j is called the semi-classical action. Our goal is to give an algorithm for computing the functions S j (E), E I. H H W E + I A + A A A+ H = H W E A K A W Figure 1: the phase space In fact exp(is h (E)/h) is the holonomy of the WKB-Maslov microlocal solutions of (Ĥ E)u = 0 around the trajectory γ E = H 1 (E) A. It is well known that: S 0 (E) = γ E ξdx = {H 0 dxdξ is the action integral E} W S 1 (E) = π γ E H 1 dt includes the Maslov correction and the subprincipal term. Our main result is: 3

4 Theorem 1 If H satisfies the previous hypothesis, we have: for j 2, S j (E) = ( ) ( 1) l 1 l 2 d P j,l (x, ξ) dt (l 1)! de γ E where 2 l L(j) t is the parametrization of γ E by the time evolution dx = (H 0 ) ξ dt, dξ = (H 0 ) x dt The P j,l s are locally (in the phase space) computable quantities: more precisely each P j,l (x, ξ) is a universal polynomial evaluated on the partial derivatives α H(x, ξ). The P j,l s are given from the Weyl symbol of the resolvent (see Proposition (1)): ( ) L(j) σ Weyl (z Ĥ) 1 = h j 1 z H 0 + j=1 l=2 P j,l (z H 0 ) l. If H = H 0, S 2j+1 (E) = 0 for j > 0. In that case, the polynomial P j,l ( α H) is homogeneous of degree l 1 w.r. to H and the total weight of the derivatives is 2j, so that all monomials in P j,l are of the form with l 1 k=1 α k = 2j and k, α k 1. Π l 1 k=1 α k H Remark 1 We have also the following nice formula 4 : for any l 2, h j P j,l (x 0, ξ 0 ) = (H H 0 (x 0, ξ 0 )) (l 1) (x 0, ξ 0 ), j where the power (l 1) is taken w.r. to the star product. Proof. Let us denote h 0 = H 0 (x 0, ξ 0 ). We have and z Ĥ = (z h 0) (Ĥ h 0) (z Ĥ) 1 = (z h 0 ) l (Ĥ h 0) l 1 l=1 The formula follows then by identification of both expressions of the Weyl symbol of the resolvent at (x 0, ξ 0 ). A less formal derivation is given by applying formula (3) to f(e) = (E h 0 ) l 1 and computing Weyl symbols at the point (x 0, ξ 0 ). 4 I learned this formula from Laurent Charles 4

5 3 Moyal formula Let us define the Moyal product a b of the semi-classical symbols a and b by the rule: Op Weyl (a) Op Weyl (b) = Op Weyl (a b) We have the well known Moyal formula (see [4]): a b = ( ) j 1 h {a, b} j j! 2i where {a, b} j (z) = [( ξ x1 x ξ1 ) j (a(z) b(z 1 ))] z1 =z with z = (x, ξ), z 1 = (x 1, ξ 1 ). In particular {a, b} 0 = ab and {a, b} 1 is the usual Poisson bracket. From the Moyal formula, we deduce the following: Proposition 1 The Weyl symbol j hj R j (z) of the resolvent (z Ĥ) 1 of Ĥ is given by L(j) h j 1 R j (z) = + h j P j,l (1) z H 0 (z H 0 ) l where the P j,l (x, ξ) are universal polynomials evaluated on the Taylor expansion of H at the point (x, ξ). If H = H 0, only even powers of j occur: R 2j = 0. Proof. The proposition follows directly from the evaluation by Moyal formula of the left-hand side of ( ) (z H) h j R j = 1. The important point is that the poles at z = H are at least of multiplicity 2 for j 1. Using ( ) ( ) (z H) h j R j = h j R j (z H) = 1, and the fact that {.,.} j are symmetric for even j s and antisymmetric for odd j s, we can prove the second statement by induction on j. 5 j=1 l=2

6 4 The method Let f C o (I) and let us compute the trace D(f) := Trace(f(ĤW )) mod O(h ) in 2 different ways: 1. Using the eigenvalues given by Bohr-Sommerfeld rules we get: Trace(f(ĤW )) = n Z f(s 1 h (2πhn)) + O(h ) and, because f S 1 h is a smooth function converging in the Co topology to f S0 1 we can apply Poisson summation formula and we get D(f) = 1 f(s 1 h 2πh (u)) du + O(h ) and D(f) = 1 f(e)s h 2πh (E) de + O(h ) R or using Schwartz distributions: R (a) D = 1 2πh S h (E) + O(h ) 2. On the other hand, we compute the Weyl symbol of f(ĥ) using Helffer- Sjöstrand s trick (see [4] p. 93): f(ĥ) = 1 F π C z=x+iy z (z)(z Ĥ) 1 dxdy (2) where F C0 (C) is a quasi-analytic extension of f, i.e. Taylor expansion 1 F (x + ζ) = k! f (k) (x)ζ k at any real x. k=0 We start with the Weyl symbol of the resolvent (1). F admits the We get then the symbol of f(ĥ) by puting Equation (1) into (2): ( [f(ĥ) = Op Weyl f(h 0 ) + ) h j (l 1)! f (l 1) (H 0 )P j,l. (3) j 1,l 2 The justification of this formal step is done in [4]. 6

7 We then compute the trace by using We get: D(f) = 1 2πh Tr (Op Weyl (a)) = 1 2πh T R ( f(h 0 ) + j 1,l 2 T R a(x, ξ) dxdξ. ) h j 1 (l 1)! f (l 1) (H 0 )P j,l dxdξ We can rewrite using dtde = dxdξ and integrating by parts: ( (b) D = 1 T(E) + ( ) l 1 ) h j ( 1)l 1 d P j,l dt 2πh (l 1)! de γ E j 1,l 2 So we get, because l 2, by identification of (a) and (b), for j 1: S j (E) ( ) ( 1) l 1 l 2 d P j,l dt = C j (4) (l 1)! de l 2 γ E where the C j s are independent of E. Proposition 2 In the previous formula (4), the C j s are also independent of the operator. Proof. We can assume that (0, 0) is in the disk whose boundary is A. Let us choose an Hamiltonian K which coïncides with H W outside the disk bounded by A and with the harmonic oscillator ˆΩ = Op Weyl ( 1 2 (x2 + ξ 2 )) near the origine. We can assume that K has no other critical values than 0. We claim: for all j 1, 1. C j ( ˆK) = C j (ˆΩ) 2. C j (Ĥ) = C j( ˆK) Both claims come from the following facts: let us give 2 Hamiltonians whose Weyl symbols coïncide in some ring B, then (i) The P j,l are the same for 2 operators in the ring B where both have the same Weyl symbol, because they are locally computed from the symbols which are the same. (ii) The S j (E) s are the same for both operators because they have the same eigenvalues in the corresponding well modulo O(h ): both operators have the same WKB-Maslov quasi-modes in B. 7

8 5 The case of the harmonic oscillator Proposition 3 For the harmonic oscillator, C 1 = π and, for j 2, C j = 0. Proof. If ˆΩ = Op Weyl ( 1 2 (x2 + ξ 2 )) is the harmonic oscillator we have: S h (E) = 2πE + πh because E n = (n 1 )h for n = 1,. 2 It remains to compute the P j,l s. Let us put ρ = 1 2 (x2 + ξ 2 ), and σ Weyl ( (z ˆΩ) 1 ) = h j R j It is clear that the R j s are functions f j (ρ, z) and from Moyal formula we get: 1 f j+2 = 4(z ρ) (f j + ρf j ) and by induction on j: f 2j+1 = 0 and f 2j (ρ, z) = l=3j+1 l=2j+1 with a j,l R. The result comes from if l 2j + 1. a l,j ρ l 2j 1 (z ρ) l, ( ) l 2 d ρ l 2j 1 dt = 0, de γ E 6 The term S 2 Let us assume first that H = H 0. From the Moyal formula, we have R 2 = 1 z H 0 {H 0, 1 } 2 = z H 0 4(z H 0 ) Γ 3 4(z H 0 ) 4 with = (H 0 ) xx (H 0 ) ξξ ((H 0 ) xξ ) 2 8

9 and Γ = (H 0 ) xx ((H 0 ) ξ ) 2 + (H 0 ) ξξ ((H 0 ) x ) 2 2(H 0 ) xξ (H 0 ) x (H 0 ) ξ. Using formulae (1) and (4), we get: S 2 (E) = 1 d 8 de Theorem 2 γ E dt ( ) 2 d Γ dt (5) de γ E If H = H 0, we have S 2 = 1 d dt (6) 24 de γ E In the general case, we have: S 2 = 1 d dt H 2 dt + 1 d H de γ E γ E 2 de dt. γ E Formula (5) were obtained in [1], formula (3.12), and formula (6) by Robert Littlejohn [6] using completely different methods. Proof. Γdt is the restriction to γ E of the 1-form α in R 2 with α = ((H 0 ) xx (H 0 ) ξ (H 0 ) xξ (H 0 ) x )dx+((h 0 ) xξ (H 0 ) ξ (H 0 ) ξξ (H 0 ) x )dξ. Orienting γ E along the Hamiltonian flow, we get using Stokes formula: Γ dt = α = dα γ E γ E D E where D E = γ E and D E is oriented by dx dξ. We have dα = 2 dx dξ and hence: Γ dt = 2 dxdξ. γ E D E From dtde = dxdξ, we get: d dxdξ = dt. de D E γ E So that: d Γ dt = 2 dt de γ E γ E from which Theorem 2 follows easily. 9

10 7 Quantum numbers Theorem 3 The quantum number n in the Bohr-Sommerfeld rules corresponds exactly to the n th eigenvalue in the corresponding well, i.e. the n th eigenvalue of ĤW. Proof. It is clear that the labelling of the eigenvalues of ĤW is invariant by homotopies leaving the symbol constant in A. We can then change Ĥ W to ˆK for which the result is clear because the quantization rules give then exactly all eigenvalues. 8 Extensions 8.1 2d phase spaces The method applies to any 2d phase space using only 3 things: The star product The fact that the trace of operators is given by (1/2πh) (the integral of their symbols) An example where you know enough to compute the C j s The power of our method is that it avoides the use of any Ansatz. Maslov contributions come only from the computation of an explicit example. 8.2 The cylinder T (R/Z) In that case, we replace the hypothesis by the following: We fix some compact intervall I = [E, E + ] R, E < E +, and we assume there exists a topological ring A, homotopic to the zero section of T (R/Z), such that A = A A + with A ± a connected component of H 1 (E ± ). We assume that H has no critical point in A We assume that A is below A + (see Figure 2). We will use the Weyl quantization for symbols which are of period 1 in x. Then Theorem 1 holds. The only change is S 1 which is now 0. The proof is the same except that the reference operator is now h i x instead of the harmonic oscillator. 10

11 A + A A Figure 2: the cylinder 8.3 Other extensions It should be nice to extend the previous method to the case of Toeplitz operators on 2-dimensional symplectic phase spaces, in the spirit of [2] and [3], and to the case of systems starting from the analysis in [5]. As remarked by Littlejohn, our method does not obviously extend to semiclassical completely integrable systems Ĥ1,, Ĥd with d 2 degrees of freedom. The reason for that is that, using the same lines, we will get only the jacobian determinant of the d BS actions which is not enough to recover the actions even up to constants. 9 Relations with KdV Let us consider the periodic Schrödinger equation Ĥ = 2 x +q(x) with q(x+1) = q(x). Let us denote by λ 1 < λ 2 λ 3 < λ 4 the eigenvalues of the periodic problem for Ĥ. Then the partition function Z(t) = n=1 e tλn admits, as t 0 +, the following asymptotic expansion Z(t) = 1 4πt ( a0 + a 1 t + + a j t j + ) + O(t ) where the a j s are of the following form a j = 1 0 A j ( q(x), q (x),, q (l) (x), ) dx 11

12 where the A j s are polynomials. The a j s are called the Korteweg-de Vries invariants because they are independent of u if q u (x) = Q(x, u) is a solution of the Korteweg-de Vries equation. See [7], [8] and [11]. Let us translate ( the ) previous objects in the semi-classical context: we have Z(h 2 ) = Tr exp( h 2 Ĥ) and h 2 Ĥ is the semi-classical operator of order 0 whose Weyl symbol is ξ 2 +h 2 q(x). If we put f(e) = e E, the partition function is exactly a trace of the form used in our method except that E e E is not compactly supported. Nevertheless, the similarity between both situations is rather clear. References [1] P.N. Argyres, The Bohr-Sommerfeld Quantization Rule and the Weyl Correspondence. Physics 2 (1965), [2] L. Charles, Berezin-Toeplitz Operators, a Semi-classical Approach. Commun. Math. Phys. 239 (2003), [3] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators. Preprint 07/02, Commun. Partial Differential Equations 28 (2003), [4] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the semi-classical limit. London Math. Soc. Lecture Notes 345, Cambridge UP (1989). [5] C. Emmrich and A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations. Commun. Math. Phys. 176, No.3 (1996), [6] R. Littlejohn, Lie Algebraic Approach to Higher Order Terms. Preprint 17p. (June 2003). [7] H.P. McKean and P. van Moerbeke, The spectrum of Hill s equation. Invent. Math. 30 (1975), [8] W. Magnus and S. Winkler, Hill s equation. New York-London-Sydney: Interscience Publishers, a division of John Wiley & Sons. VIII, 127 p. (1966). [9] A. Voros, Développements semi-classiques. Thèse (Orsay, 1977). [10] A. Voros, Asymptotic -expansions of stationary quantum states. Ann. Inst. H. Poincaré Sect. A (N.S.) 26 (1977), [11] V. Zakharov and L. Faddeev, Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl. 5 (1977),